Fourier Theory in Seismic Processing

(From Liner and Ikelle and Amundsen)

• Temporal aliasing• Spatial aliasing

Sample Rates

What is the fewest number of times I need to sample this waveform per second?

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Sample Rates

Sample Rates

Sample Rates

Sample Rates

What is the fewest number of times I need to sample this waveform per second?

At least twice per wavelength or period!

OTHERWISE ….

Undersampled waveforms

True frequency (f -true)

Am

plit

ud

e

Reconstructed frequency

(f -aliased)

ff

Oversampled waveforms

= True frequency (f -true)

Am

plit

ud

e

Reconstructed frequency

frequency is unaliased

Nyquist frequency

Nyquist frequency = 1 / twice the sampling rate

Minimum sampling rate must be at least twice the desired frequency

E.g., 1000 samples per second for 500Hz,

2000 samples per second for 1000 Hz

Oversampled waveformsA

mp

litu

de Nyquist frequency

In practice we are best oversampling by double the required minimum

i.e. 1000 samples per second for a maximum of 500 Hz

i.e., 2000 samples per second for a maximum of 1000 Hz

Oversampling is relatively cheap.

Spatial frequency, or wavenumber (k) is the number of cycles per unit distance.

One spatial cycle or wavenumber = frequency/velocity.

Each wavenumber must be sampled at least twice per wavelength

(two CMP’s per wavelength)

Spatial aliasing

1

2( )kN CMPspacing

IN PRACTICE each wavenumber must be sampled at least four times per minimum

wavelength (two CMP’s per wavelength)

Spatial aliasing

However, dip (theta) as well as frequency and velocity event changes the number of cycles per distance, so

4sin

lambdaCMPinterval

Liner, 9.7,p.192

Spatial aliasing

4sin

lambdaCMPinterval

x

V t

limitsinV t

x

For aliasing NOT to occur, delta(t) must be less than T/2

Spatial aliasing

limitsin2

VT

x

lim 2sinit

VTx

Geophone Spacing and Spatial Aliasing

K=0

1/4 wavelength shift per trace

total shift across array=3/4 wavelength

K=+ or -ve?

1/4 wavelength shift per trace

total shift across array=3/4 wavelength

K=?

1/2 wavelength shift per trace

total shift across array=3/2 wavelength

K=0

3/4 wavelength shift per trace

total shift across array=2 1/4 wavelength